STAT 430/510 Probability Hui Nie Lecture 1 May 26th, 2009
Introduction Probability is the study of randomness and uncertainty. In the early days, probability was associated with games of chance, such as gambling. It is useful to have an effective method for counting the number of ways that things can happen.
A fair coin is rolled twice. You win $100 if head happens twice. What is the probability that you win? Solution: 0.25. Why? A fair coin is rolled n times. You win $100 if head happens all the time. What is the probability that you win?
The Basic Principle of Counting The Basic Principle of Counting Suppose that two experiments are to be performed. Then if experiment 1 can result in anyone of m possible outcomes and if, for each outcome of experiment 1, there are n possible outcomes of experiment 2, then together there are mn possible outcomes of the two experiments.
A small community consists of 10 women, each of whom has 3 children. If one woman and one of her children are to be chosen as mother and child of the year, how many different choice are possible? Solution: 10 3 = 30.
The Generalized Basic Principle of Counting The Generalized Basic Principle of Counting If r experiments that are to be performed are such that the first one may result in any of n 1 possible outcomes; and if, for each of these n 1 possible outcomes, there are n 2 possible outcomes of the second experiment; and if, for each of the possible outcomes of the first two experiments, there are n 3 possible outcomes of the third experiment; and if..., then there is a total of n 1 n 2 n r possible outcomes of the r experiments.
How many different 7-place license plates are possible if the first 3 places are to be occupied by letters and the final 4 by numbers? Solution: 26 26 26 10 10 10 10 = 175760000
when repetition among letters or numbers were prohibited, how many different 7-place license plates are possible if the first 3 places are to be occupied by letters and the final 4 by numbers? Solution: 26 25 24 10 9 8 7 = 78624000
Permutations An ordered sequence of k distinct objects taken from a set of n objects is called a permutation of size k. The total number of permutations of size k from n objects is denoted by P k,n. P k,n = n(n 1)(n 2) (n k + 1) = In particular, P n,n = n! n! (n k)!
How many different batting orders are possible for a baseball team consisting of 9 players? Solution: 9! = 362880
Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematical books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms Jones wants to arrange her books so that all the books dealing with the same subjects are together on the shelf. How many different arrangements are possible? Solution: 4!4!3!2!1! = 6912
How many different letter arrangements can be formed from the letters PEPPER? Solution: 6! 3!2!1! = 60
Permutations: Continued In general, suppose that we have n objects, of which n 1 are alike, n 2 are alike,..., n k are alike. How many different permutations of the n objects are there? The total number of permutations of the n objects is n! n 1!n 2! n r!
How many different signals, each consisting of 9 flags hung in a line, can be made from a set of 4 white flags, 3 red flags, and 2 blue flags if all flags of the same color are identical? Solution: 9! 4!3!2! = 1260